高阶流匹配理论证明了其在分布估计中的最优性，为生成模型提供了更优轨迹优化保障。
Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows
- 通过高阶微分方程建模加速轨迹优化，提升生成路径精度。
- 第二阶流匹配的误差上界与目标分布光滑度呈多项式关系。
- 适用于追求高精度生成的学者，尤其关注理论保证的研究者。
流匹配已成为一种强大的生成建模框架，相比扩散模型具有计算优势，因其采用确定性常微分方程（ODE）而非随机动力学。尽管已有工作证明标准流匹配在Wasserstein距离下的最坏情况最优性，但高阶流匹配——通过引入加速度项以优化样本轨迹——的理论保证仍未知。本文填补这一空白，证明高阶流匹配保持作为分布估计器的最坏情况最优性。我们推导出第二阶流匹配的估计误差上界，表明收敛速率依赖于目标分布的光滑度（以Besov空间衡量）及ODE动态的关键参数。分析中使用具有精细控制深度、宽度和稀疏性的神经网络近似，有效约束了小时间区间与大时间区间上的加速度误差，最终将结果统一为所有时间步的一般最坏情况最优界。
原文摘要 · Abstract (English)
Flow matching has emerged as a powerful framework for generative modeling, offering computational advantages over diffusion models by leveraging deterministic Ordinary Differential Equations (ODEs) instead of stochastic dynamics. While prior work established the worst case optimality of standard flow matching under Wasserstein distances, the theoretical guarantees for higher-order flow matching - which incorporates acceleration terms to refine sample trajectories - remain unexplored. In this paper, we bridge this gap by proving that higher-order flow matching preserves worst case optimality as a distribution estimator. We derive upper bounds on the estimation error for second-order flow matching, demonstrating that the convergence rates depend polynomially on the smoothness of the target distribution (quantified via Besov spaces) and key parameters of the ODE dynamics. Our analysis employs neural network approximations with carefully controlled depth, width, and sparsity to bound acceleration errors across both small and large time intervals, ultimately unifying these results into a general worst case optimal bound for all time steps.
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